PREFACE ..................................................... xxiii
PRÉCIS .......................................................... 1
1 GEOMETRY ................................................... 9
1.1 NOTATION .................................................. 11
1.1.1 Contraction ........................................ 12
1.1.2 Scalar and Dyadic Products of Complex-valued
Vectors ............................................ 12
1.2 SPECTRAL REPRESENTATION OF SECOND-ORDER TENSORS ........... 14
1.2.1 Coordinate Transformation .......................... 14
1.2.2 Eigenvalues and Eigenvectors ....................... 15
1.2.3 Symmetric Tensors .................................. 15
1.2.4 Nonsymmetric Tensors ............................... 16
1.2.5 Complex-valued Eigenvectors ........................ 18
1.3 REPRESENTATION OF TENSORS IN OBLIQUE COORDINATE TRIADS .... 18
1.3.1 Reciprocal Base Vectors ............................ 19
1.3.2 Real Eigenvalues and Eigenvectors .................. 20
1.3.3 Spectral Representation of Am ...................... 22
1.3.4 Distinct Real Eigenvalues .......................... 23
1.3.5 Two Repeated Real Eigenvalues ...................... 23
1.3.6 Three Repeated Real Eigenvalues .................... 24
1.3.7 Complex-valued Eigenvalues and Eigenvectors ........ 25
1.4 IDENTITIES FOR SECOND-ORDER TENSORS ....................... 27
1.4.1 Rivlin's Identities ................................ 27
1.4.2 Other Related Identities ........................... 28
1.4.3 Solution of AX + XA = B, Given A and В ............. 29
1.4.4 Solution of AX - XA = B, Given A and В ............. 31
A with Distinct Eigenvalues ........................ 31
A with Repeated Eigenvalues ........................ 32
1.5 ISOTROPIC TENSOR-VALUED FUNCTIONS ......................... 34
1.5.1 A Class of Isotropic Tensor-valued Functions of
Real-valued Second-order Tensors ................... 35
1.5.2 A with Distinct Real Eigenvalues ................... 36
1.5.3 A with Distinct Complex Eigenvalues ................ 37
1.5.4 A with Two Repeated Eigenvalues .................... 37
1.5.5 A with Three Repeated Eigenvalues .................. 38
1.6 DERIVATIVE OF ISOTROPIC TENSOR-VALUED FUNCTION ............ 39
1.6.1 Derivative of a Second-order Tensor ................ 40
1.6.2 Coordinate-independent Representation .............. 41
1.6.3 Representation of f(A) ............................. 41
Distinct Eigenvalues ............................... 42
Two Repeated Eigenvalues ........................... 42
Three Repeated Eigenvalues ......................... 43
1.6.4 Expression for g{A, HA(Ǻ)} ......................... 44
1.7 REFERENCES ................................................ 47
2 KINEMATICS ................................................ 49
2.1 MOTION AND DEFORMATION OF A CONTINUUM ..................... 51
2.1.1 Deformation Gradient ............................... 52
2.1.2 Principal Stretches ................................ 53
2.2 POLAR DECOMPOSITION ....................................... 55
2.2.1 Spectral Representation ............................ 55
2.2.2 Basic Invariants and Direct Evaluation of U, R,
and V .............................................. 56
2.2.3 Lagrangian and Eulerian Triads ..................... 59
2.2.4 Transformation of Volume Elements .................. 61
2.2.5 Transformation of Surface Elements ................. 62
2.3 STRAIN MEASURES ........................................... 63
2.3.1 Explicit Representation of General Strain
Measures in Terms of U and V ....................... 63
Distinct Principal Stretches ....................... 64
Two Repeated Principal Stretches ................... 64
Three Repeated Principal Stretches ................. 65
2.3.2 Required Properties of Strain Measures ............. 66
2.3.3 Lagrangian and Eulerian Strain Tensors ............. 66
2.3.4 Biot's Strain Measure .............................. 66
2.3.5 Logarithmic Strain Measure ......................... 66
2.3.6 Other Strain Measures .............................. 67
2.4 VELOCITY AND ACCELERATION ................................. 68
2.4.1 Velocity ........................................... 68
2.4.2 Acceleration ....................................... 68
2.5 DEFORMATION-RATE AND SPIN TENSORS ......................... 69
2.5.1 Deformation-rate and Spin Tensors .................. 69
2.5.2 Stretching, Shearing, and Volumetric Deformation
Rate ............................................... 70
2.6 OTHER SPIN TENSORS ........................................ 71
2.7 RELATIONS BETWEEN RIGHT STRETCH RATE AND DEFORMATION
RATE ...................................................... 72
2.7.1 Alternative Coordinate-independent Expressions ...... 74
2.8 RELATIONS BETWEEN LEFT STRETCH RATE AND DEFORMATION RATE .. 76
2.8.1 Alternative Coordinate-independent Expressions ...... 77
2.9 RELATIONS AMONG VARIOUS SPIN TENSORS ...................... 79
2.10 STRAIN RATES .............................................. 82
2.10.1 Strain Rates in Lagrangian Triad ................... 82
2.10.2 Strain Rates in Eulerian Triad ..................... 84
2.10.3 Coordinate-independent Expressions ................. 84
2.10.4 Summary of Coordinate-independent Expressions for
General Strain Rates ............................... 86
Distinct Eigenvalues ...................................... 86
Two Repeated Eigenvalues .................................. 87
Three Repeated Eigenvalues ................................ 87
2.11 REFERENCES ................................................ 87
3 STRESS AND STRESS-RATE MEASURES, AND BALANCE RELATIONS .... 89
3.1 STRESS MEASURES ........................................... 91
3.1.1 Balance of Angular and Linear Momentum ............. 91
3.1.2 Second Piola-Kirchhoff Stress ...................... 92
3.1.3 General Relations in Principal Coordinates ......... 93
3.1.4 General Coordinate-independent Relations ........... 93
Distinct Principal Strains ......................... 95
Two Repeated Principal Strains ..................... 95
Three Repeated Principal Strains ................... 95
3.1.5 First-order Accurate Expressions ................... 96
3.1.6 Stress Conjugate to Lagrangian Strain Measure ...... 97
3.1.7 Stress Conjugate to Biot's Strain Measure .......... 97
3.1.8 Stress Conjugate to Logarithmic Strain Measure ..... 97
3.1.9 Stresses Conjugate to Other Strain Measures ........ 98
3.1.10 Nominal Stress ..................................... 98
3.1.11 Summary of Relations Among Various Stress
Measures ........................................... 99
3.2 PHYSICAL RELATIONS AMONG COMMONLY USED STRESS MEASURES .... 99
3.2.1 Contravariant Components of Kirchhoff Stress ...... 101
3.2.2 Cartesian-contravariant Components of Kirchhoff
Stress ............................................ 102
3.2.3 Covariant-contravariant Components of Kirchhoff
Stress ............................................ 103
3.2.4 Covariant Components of Kirchhoff Stress .......... 103
3.2.5 Cartesian-covariant Components of Kirchhoff
Stress ............................................ 104
3.3 STRESS-RATE MEASURES ..................................... 105
3.3.1 Jaumann Stress Rate ............................... 105
3.3.2 Convected Stress Rate ............................. 105
3.4 GENERAL RESULTS ON OBIECTIVE STRESS RATES ................ 107
3.4.1 General Relation Connecting Objective Stress
Rates ............................................. 107
3.5 STRESS RATES WITH CURRENT STATE AS REFERENCE ............. 109
3.5.1 Nominal Stress Rate ................................ 109
3.6 BALANCE RELATIONS AND BOUNDARY-VALUE PROBLEMS ............ 110
3.6.1 Conservation of Mass .............................. 110
3.6.2 Balance of Linear Momentum ........................ 110
3.6.3 Balance of Angular Momentum ....................... 111
3.6.4 Conservation of Energy ............................ 111
3.6.5 Boundary-Value Problems ........................... 111
3.7 PRINCIPLE OF VIRTUAL WORK ................................ 112
3.8 FINITE-ELEMENT FORMULATION ............................... 113
3.8.1 Weak Form of Equations of Motion in the Current
Configuration ..................................... 114
3.8.2 Weak Form of the Rate Equations of Motion ......... 115
3.8.3 Discretized Equations of Motion ................... 117
3.9 BALANCE RELATIONS WITH SURFACES OF DISCONTINUITY ......... 118
3.9.1 Kinematics of Volume Integrals .................... 118
3.9.2 Transport Theorem ................................. 119
3.9.3 Dynamical Conditions at Surfaces of
Discontinuity ..................................... 119
3.9.4 Balance Relations at Surfaces of Discontinuity .... 121
3.10 GENERAL VARIATIONAL PRINCIPLES FOR HYPERELASTIC
MATERIALS ................................................ 122
3.10.1 Preliminaries ..................................... 124
3.10.2 General Variational Principles .................... 125
3.10.3 General Variational Principle in Terms of Nominal
Stress and Deformation Gradient ................... 127
3.10.4 Variational Principle with Two Independent
Fields ............................................ 128
Strain-energy Functional .......................... 128
Complementary-energy Functional ................... 130
3.10.5 Variational Principles with Discontinuous Fields .. 132
3.10.6 Incremental Formulation ........................... 133
3.10.7 Linearized Formulation ............................ 137
3.11 REFERENCES ............................................... 138
4 CONTINUUM THEORIES OF ELASTOPLASTICITY ................... 143
4.1 ELASTIC AND INELASTIC POTENTIALS ......................... 145
4.1.1 Elastic Potentials ................................ 145
4.1.2 Inelastic Potential and Normality Rule ............ 148
4.1.3 Normality Rules ................................... 149
4.1.4 On the Existence of the Inelastic Potential ....... 150
4.2 RATE-INDEPENDENT THEORIES ................................ 152
4.2.1 Yield Surface ..................................... 153
4.2.2 Constitutive Relations: Smooth Yield Surface ...... 153
4.2.3 Yield Surface in Stress Space ..................... 153
4.2.4 Yield Surface in Strain Space ..................... 155
4.2.5 Flow Potential and Associative Flow Rule .......... 157
4.2.6 The J2-flow Theory with Isotropic Hardening ...... 158
4.2.7 The J2-flow Theory with Kinematic Hardening ...... 160
4.2.8 The J2-flow Theory with Combined Isotropic-
kinematic Hardening ............................... 163
4.3 DILATANCY AND PRESSURE SENSITIVITY ....................... 163
4.3.1 Nonassociative Plasticity with Dilatancy and
Friction .......................................... 165
4.3.2 Application: Triaxial Test of Soils ............... 168
4.3.3 A Model for Cohesionless Sands .................... 171
4.3.4 Model with Cohesion ............................... 176
4.3.5 Simple Shear ...................................... 177
4.3.6 Comparison with Experimental Results .............. 177
4.3.7 Application to Porous Metals ...................... 181
4.4 YIELD VERTEX MODELS ...................................... 183
4.4.1 Strain-space Representation ....................... 184
4.4.2 Stress-space Representation ....................... 184
4.4.3 Consistency Conditions ............................ 185
4.5 NONLINEAR ELASTICITY AND DEFORMATION THEORIES ............ 185
4.5.1 Isotropic Elasticity .............................. 186
4.5.2 The J2-deformation Theory ......................... 187
4.5.3 Generalization to Three-dimensional Vertex Model .. 188
4.5.4 Relation to Granular-material Models and Summary
of Equations ...................................... 190
4.6 THE J2-FLOW THEORY WITH NONCOAXIALITY ................... 191
4.6.1 Nondilatant Model with Noncoaxiality ............... 191
4.7 MODELS FOR FRICTIONAL GRANULAR MATERIALS ................. 192
4.7.1 Backstress, Fabric, and Dilatancy ................. 196
4.7.2 Yield Criterion ................................... 198
4.7.3 Plastic Strain Rate with Dilatancy and
Noncoaxiality ..................................... 199
4.7.4 Energy Balance Equation ........................... 200
4.7.5 Dilatancy, Friction, and Fabric ................... 201
4.7.6 Evolution Equations ............................... 203
4.7.7 Elasticity Relations .............................. 206
4.7.8 General Constitutive Relations .................... 208
4.7.9 The Case of Isotropic Elasticity .................. 210
4.7.10 Granular Materials with Vanishing Elastic Range
in Shearing ....................................... 210
4.8 RATE-DEPENDENT THEORIES .................................. 212
4.8.1 Flow Stress: Empirical Models ..................... 214
4.8.2 Physically-based Models ........................... 217
4.8.3 Thermal Activation and Flow Stress ................ 219
4.8.4 A Simple Model .................................... 221
4.8.5 More General Models ............................... 222
4.8.6 Dislocations as Short-range Barriers .............. 222
4.8.7 Effect of Long-range Barriers ..................... 226
4.8.8 Drag-controlled Plastic Flow ...................... 227
4.8.9 Application: Flow Stress of Commercially Pure
Tantalum .......................................... 229
Data Analysis ..................................... 231
Effect of Long-range Barriers ..................... 231
Comparison with Data used for Modeling ............ 233
Discussion of the Model ........................... 234
Application to Molybdenum ......................... 235
4.8.10 Application to OFHC Copper ........................ 235
Experimental Data ................................. 237
Effect of Long-range Barriers ..................... 237
Other Metals ...................................... 241
4.8.11 Rate-dependent J2-plasticity ...................... 242
4.8.12 Viscoplastic J2-flow Theory ....................... 245
4.8.13 Viscoplastic J2-flow Theory with Noncoaxiality .... 246
4.9 GENERAL ANISOTROPIC ELASTOPLASTICITY ..................... 246
4.9.1 Decomposition of Deformation Gradient ............. 250
4.9.2 Relations Among Rate Quantities ................... 252
4.9.3 Explicit Expressions for Objective Spin Tensors
Ŵe, Ŵp, We, and Wp ................................ 255
4.9.4 Material Frame Indifference ....................... 258
4.9.5 Elastic Response .................................. 259
4.9.6 Comments on Elastic Anisotropy .................... 262
4.9.7 Calculation of Stress Rate ........................ 266
4.9.8 Small Elastic Deformations ........................ 267
4.9.9 Evolution of Back Stress .......................... 269
4.9.10 Alternative Decomposition of Deformation
Gradient .......................................... 269
4.10 REFERENCES ............................................... 270
5 INTEGRATION OF CONTINUUM CONSTITUTIVE
EQUATIONS ................................................ 285
5.1 INTRODUCTION ............................................. 287
5.2 INCREMENTAL KINEMATICS ................................... 288
5.2.1 Constant Velocity Gradient ........................ 289
5.2.2 Unidirectional Stretch ............................ 290
5.2.3 Exact Coordinate-independent Relations Between
Velocity and Incremental Deformation Gradients .... 293
Constant Velocity Gradient ........................ 293
Unidirectional Stretch ............................ 294
5.3 I2-FLOW THEORIES ......................................... 296
5.3.1 Rate-independent Model ............................ 297
5.3.2 Rate-dependent Model .............................. 297
5.3.3 Thermal Softening with Isotropic Hardening ........ 298
5.4 PLASTIC-PREDICTOR/ELASTIC-CORRECTOR METHOD WITH ERROR
ESTIMATE ................................................. 300
5.4.1 Stiff Systems ..................................... 301
5.4.2 The Constitutive Algorithm ........................ 301
(A) Rate-independent elastoplasticity ............. 303
(B) Rate-dependent elastoviscoplasticity .......... 304
5.4.3 Error Estimate .................................... 304
Case I: Rate-independent elastoplasticity ......... 306
Case II: Rate-dependent elastoviscoplasticity ..... 306
5.5 SINGULAR PERTURBATION METHOD FOR CONSTITUTIVE
ALGORITHMS ............................................... 308
5.5.1 Singular Perturbation Solution .................... 309
5.5.2 Rate-independent Elastoplasticity ................. 311
5.5.3 Rate-dependent Elastoviscoplasticity .............. 312
5.5.4 Modified Outer Solution Method .................... 314
5.6 PROPORTIONAL LOADING ..................................... 315
5.6.1 Rate-independent Model ............................ 316
Forward-gradient Method ........................... 316
Iterative Method .................................. 317
Plastic-predictor/Elastic-corrector Method ........ 317
5.6.2 Rate-dependent Model .............................. 319
Tangent-modulus Method ............................ 319
Iterative Method .................................. 320
Plastic-predictor/Elastic-corrector Method ........ 321
Computational Steps in Plastic-predictor/
Elastic-corrector Method .......................... 326
5.6.3 Asymptotic Analysis ............................... 327
5.6.4 Reverse Loading ................................... 328
Rate-independent Model ............................ 328
Rate-dependent Model .............................. 329
5.7 INTEGRATION FOR UNIDIRECTIONAL STRETCH ................... 330
5.7.1 Elastic, Perfectly-plastic Model .................. 331
Radial-return Method .............................. 331
Exact Integration ................................. 332
5.7.2 Rate-independent Model ............................ 333
Generalized Radial-return Method .................. 333
Computational Steps ............................... 334
Example ........................................... 335
Perfect-plasticity Path ........................... 337
Computational Steps ............................... 338
5.7.3 Rate-dependent Model .............................. 339
Generalized Radial-return Method .................. 340
Computational Steps ............................... 341
Example ........................................... 343
Perfect-plasticity Path ........................... 343
Computational Steps ............................... 346
5.7.4 Elasticity-dominated Deformation .................. 348
Rate-independent Model ............................ 349
Rate-dependent Model .............................. 350
(i) Strain-rate softening ......................... 351
(ii) Strain-rate hardening and unloading .......... 352
5.8 INTEGRATION FOR CONSTANT VELOCITY GRADIENT ............... 353
5.8.1 Elastic, Perfectly-plastic Model .................. 353
Generalized Radial-return Method .................. 353
Approximate Integration ........................... 356
Exact Integration ................................ 358
Approximate but Effective Integration Procedure ... 361
5.8.2 Rate-independent Model ............................ 362
Computational Steps for Generalized Radial-
return Method ..................................... 362
Computational Steps for the Perfect-plasticity
Path Method ....................................... 364
5.8.3 Rate-dependent Model .............................. 367
Computational Steps for Generalized Radial-
return Method ..................................... 367
Computational Steps for the Perfect-plasticity
Path Method ....................................... 369
5.9 REFERENCES .............................................. 373
APPENDIX 5.A IDENTITIES FOR SECOND-ORDER DEVIATORIC
AND SKEWSYMMETRIC TENSORS ................................ 375
APPENDIX 5.B SOLUTION OF A + θ(AW - WA) = В .............. 377
6 FINITE ELASTOPLASTIC DEFORMATION OF SINGLE CRYSTALS ...... 381
6.1 PHYSICS OF CRYSTAL PLASTICITY ............................ 383
6.1.1 Crystal Structure and Elasticity .................. 383
6.1.2 Plasticity and Slip ............................... 385
6.1.3 Dislocations ...................................... 386
6.1.4 Burgers'Vector .................................... 387
6.1.5 Action of a Stress Field on a Dislocation ......... 390
6.1.6 Elastic Field of a Dislocation .................... 393
Screw Dislocation ................................. 393
Edge Dislocation .................................. 395
A Dislocation in an Anisotropic Solid ............. 397
6.1.7 Slip Systems ...................................... 400
6.1.8 Slip Systems in fee and bec Crystals .............. 400
6.1.9 Dislocation-induced Distortion .................... 401
6.1.10 Dislocation Motion and Plastic Distortion Rate .... 401
6.2 KINEMATICS OF FINITE DEFORMATION OF SINGLE CRYSTALS ...... 403
6.2.1 Decomposition of Deformation Gradient ............. 404
6.2.2 Decomposition of Velocity Gradient ................ 407
6.2.3 Plastic Distortion ................................ 411
6.2.4 Elastic Lattice Distortion Rate and Spin .......... 413
6.2.5 Small Lattice Distortion .......................... 415
6.3 CONSTITUTIVE EQUATIONS FOR SINGLE CRYSTALS ............... 415
6.3.1 Crystal Elasticity ................................ 416
6.3.2 Crystal Elastoplasticity Relative to Current
Configuration ..................................... 417
6.3.3 Crystal Elastoplasticity with General Strain and
Stress Measures ................................... 422
6.3.4 Crystal Elastoplasticity Relative to Elastically
Relaxed Lattice ................................... 422
6.3.5 Crystal Elastoplasticity Relative to Undeformed
Configuration ..................................... 424
6.3.6 Rate of Stress Work and Measure-invariance of
Plastic Dissipation ............................... 425
6.3.7 Crystal Elastoplasticity with Small Elastic
Strains ........................................... 426
6.4 RESISTANCE TO SLIP AND WORKHARDENING: RATE-INDEPENDENT
MODELS ................................................... 427
6.4.1 Schmid Law ......................................... 427
6.4.2 Yield Surface and Plastic Normality in Stress
Space ............................................. 429
6.4.3 Yield Surface and Plastic Normality in Strain
Space ............................................. 432
6.4.4 Slip Rates ........................................ 433
6.4.5 Critical Shear Stress and Hardening Matrix ........ 434
6.4.6 Latent Hardening .................................. 436
6.4.7 Non-Schmid Effects ................................ 440
6.5 PHYSICALLY-BASED SLIP MODELS: RATE AND TEMPERATURE
EFFECTS .................................................. 440
6.5.1 Long-range Barriers and Latent Hardening .......... 442
6.5.2 Self Hardening of a Previously Latent System ...... 445
6.5.3 Resistance to Crystalline Slip: Long-range
Barriers .......................................... 447
6.5.4 Resistance to Crystalline Slip: Short-range
Barriers .......................................... 448
Slip Rate in bcc Crystals ......................... 448
Slip Rate in fee Crystals ......................... 450
6.5.5 Effect of Viscous Drag ............................ 451
6.5.6 bcc Crystals ...................................... 452
Interdependency of Slip Systems ................... 452
6.5.7 fee Crystals ...................................... 453
6.5.8 Schmid Rule ....................................... 457
6.5.9 Fully Developed Plastic Flow ...................... 457
Eight Active Slip Systems ......................... 458
Six Active Slip Systems ........................... 458
6.5.10 Numerical Simulation of Single Crystals ........... 461
Kinematical Relations ............................. 461
Calculation of Temperature ........................ 462
6.5.11 Computational Algorithm ........................... 462
Plastic Predictor and Transition Regime ........... 464
Elastic Corrector ................................. 465
Lattice Rotation .................................. 466
Resolved Shear Stresses of Inactive Slip Systems .. 467
Rapidly Changing Regime ........................... 467
Fully Developed Flow .............................. 467
6.5.12 Power-law Model for fee Crystals .................. 468
Example, fee Crystal .............................. 468
6.5.13 Dynamic Collapse of Single-crystal Hollow
Cylinder .......................................... 469
6.5.14 Dislocation-based Model for bcc and fee Crystals .. 471
Example, bcc Crystal .............................. 473
6.5.15 bcc and fee Polycrystals .......................... 473
Polycrystal Calculations .......................... 474
Simulation of Response of Tantalum ................ 475
6.5.16 Numerical Simulation of Polycrystal Copper ........ 476
6.5.17 Length Scales and Size Effect in Crystal
Plasticity ........................................ 476
Dislocations and Length Scales .................... 480
A Simple Couple-stress Theory ..................... 483
Microstructure and Continuously Distributed
Dislocations ...................................... 486
Phenomenological Strain-gradient Plasticity ....... 490
6.6 REFERENCES ............................................... 491
APPENDIX 6.A MILLER INDICES .............................. 498
Miller Indices for Directions ............................ 499
Miller Indices for Planes ................................ 500
hep Crystals ............................................. 500
Zone Axis ................................................ 501
7 FINITE PLASTIC DEFORMATION OF GRANULAR MATERIALS ......... 503
7.1 INELASTIC DEFORMATION OF GRANULAR MASSES ................. 505
7.1.1 Dilatancy, Densification, and Fabric in Simple
Shearing .......................................... 506
7.1.2 Relation to Continuum Models ...................... 513
7.2 STRESS MEASURES IN GRANULAR MASSES ....................... 514
7.2.1 Overall Cauchy Stress in Granular Media ........... 515
Theorem I ......................................... 515
Contact Forces and Branch Vectors ................. 517
Theorem II ........................................ 518
Direct Proof ...................................... 518
Proof by Virtual-work Method ...................... 519
7.2.2 Other Overall Stress Measures in Granular Media ... 519
7.2.3 Classification of Contacts ........................ 520
Distribution Density Function of Branch
Orientations ...................................... 521
7.2.4 Overall Stress Tensor and Average Traction
Vectors ........................................... 522
Distribution Density Function of Branch
Intersections with a Plane ........................ 523
7.2.5 Symmetry of Overall Cauchy Stress Tensor .......... 525
7.2.6 Stress-fabric Relations ........................... 525
7.2.7 Stress-fabric Relations for Spherical Granules .... 526
7.2.8 Stress-fabric Relations for Nonspherical
Granules .......................................... 529
7.2.9 Nominal Stress Tensor ............................. 532
7.2.10 Stress-rate Measures .............................. 533
7.3 FABRIC MEASURES IN AN ASSEMBLY OF RIGID GRANULES ......... 533
7.3.1 Scalar Measures ................................... 534
Void Ratio and Porosity ........................... 534
Density of Grains ................................. 534
Average Coordination Number ....................... 535
Density of Contacts ............................... 535
Average Branch Length ............................. 536
7.3.2 Vector Measures ................................... 537
Density of Branch Intersections with a Plane ...... 537
Angular Distribution of Branches Intersecting
the ν-plane ....................................... 539
Solid Paths ....................................... 539
7.3.3 Fabric Tensors .................................... 541
Angular Distribution of Branches and Contact
Normals ........................................... 541
7.3.4 Other Fabric Measures ............................. 543
7.4 EXPERIMENTAL EVALUATION OF FABRIC-STRESS RELATIONS ....... 544
7.4.1 Photoelastic Granules ............................. 546
7.4.2 Biaxial Experiments ............................... 547
Apparatus ......................................... 547
Sample Preparation ................................ 547
Test Results ...................................... 549
Evolution of Distribution of Contact Normals ...... 551
7.4.3 Simple Shearing Experiments ....................... 554
Apparatus ......................................... 554
Sample Preparation and Test Procedure ............. 556
Stress-microstructure Relation .................... 558
Mechanism of Strain Hardening ..................... 559
Dilatancy ......................................... 560
Representation of Distribution Density Functions .. 561
7.5 MICROMECHANICALLY-BASED CONSTITUTIVE MODELS FOR
FRICTIONAL DEFORMATION OF GRANULAR MASSES ................ 564
7.5.1 Previous Work ..................................... 564
7.5.2 Model Assumptions ................................. 567
7.5.3 Resistance to Sliding ............................. 568
7.5.4 A Two-dimensional Model ........................... 570
7.5.5 Meso-scale Yield Condition ........................ 573
7.5.6 Loading and Unloading ............................. 576
7.5.7 A Rate-independent Double-sliding Model ........... 577
7.5.8 Deformation Rate and Spin, Based on Double
Sliding ........................................... 579
7.5.9 Elasticity Relations .............................. 581
7.5.10 Fabric and its Evolution .......................... 582
7.5.11 Continuum Approximation ........................... 583
7.5.12 Void Ratio ........................................ 583
7.5.13 Consistency Condition ............................. 584
7.5.14 Dilatancy ......................................... 585
7.5.15 Material Parameters ............................... 585
7.5.16 Example ........................................... 586
7.5.17 Generalization to Three Dimensions ................ 588
Stress Tensor ..................................... 588
Stress-rate Tensor ................................ 588
Fabric and its Evolution .......................... 589
Yield and Consistency Conditions .................. 589
Dilatancy ......................................... 589
7.6 REFERENCES ............................................... 590
8 AVERAGE QUANTITIES AND HOMOGENIZATION MODELS ............. 595
8.1 AVERAGING THEOREMS ....................................... 597
8.1.1 Comments on Continuum Length Scale ................ 597
8.1.2 Choice of Deformation and Stress Measures ......... 598
Notation .......................................... 598
8.1.3 Average Deformation and Deformation-rate
Measures .......................................... 599
8.1.4 Average Stress Measures ........................... 600
8.1.5 Average Stress-rate Measures ...................... 601
8.1.6 General Identities ................................ 601
8.1.7 Uniform Boundary Tractions and Traction Rates ..... 603
8.1.8 Uniform Boundary Displacements and Displacement
Rates ............................................. 603
8.1.9 General Identities for Uniform Boundary Data ...... 604
8.2 HOMOGENIZATION AND CONCENTRATION TENSORS ................. 604
8.2.1 Eshelby's Tensor .................................. 605
8.2.2 General Phase-Transformation Problem .............. 606
8.2.3 Homogenization and Eigenvelocity Gradient ......... 607
8.2.4 Eigenstress Rate .................................. 610
8.2.5 Consistency Conditions ............................ 611
8.2.6 Concentration Tensors ............................. 613
8.3 THE GREEN FUNCTION AND CONCENTRATION TENSORS ............. 614
8.3.1 Reciprocal Relations .............................. 615
8.3.2 Green's Function .................................. 616
8.3.3 The Body-force-rate Problem ....................... 617
8.3.4 The Eigenvelocity Gradient or Eigenstress Rate
Problem ........................................... 619
8.3.5 Generalized Eshelby Tensor and its Conjugate ...... 621
8.4 AVERAGE QUANTITIES ....................................... 622
8.4.1 Double-inclusion Problem .......................... 623
8.4.2 Generalized Double-inclusion Problem .............. 626
8.4.3 A Nested Sequence of Transforming Ellipsoidal
Regions ........................................... 628
8.5 CALCULATION OF THE GREEN FUNCTION ........................ 630
8.5.1 Strong Ellipticity and Green's Function ........... 631
8.5.2 Green's Function .................................. 632
8.5.3 Generalized Eshelby Tensor and its Conjugate ...... 634
8.6 AVERAGING MODELS ......................................... 637
8.6.1 Consistency Restrictions for Concentration
Tensors ........................................... 638
8.6.2 Voigt and Reuss Models ............................ 640
8.6.3 Dilute-distribution Model ......................... 640
8.6.4 Self-consistent Model ............................. 640
8.6.5 Double-inclusion Model ............................ 641
8.6.6 Reduction to Self-consistent Model ................ 643
8.6.7 Two-phase Model ................................... 643
8.6.8 Overall Pseudo-compliance Tensor .................. 644
8.6.9 Multi-inclusion Model ............................. 644
8.7 POLYCRYSTALS ............................................. 646
8.7.1 Overall Moduli for Polycrystals ................... 647
8.7.2 Self-consistent Model ............................. 648
8.7.3 Choice of Reference State ......................... 648
8.7.4 Updated Lagrangian Calculation .................... 650
8.7.5 Rate-dependent Models ............................. 650
8.8 REFERENCES ............................................... 652
APPENDIX 8.A CALCULATION OF THE ESHELBY TENSOR IN TWO
DIMENSIONS ............................................... 656
9 SPECIAL EXPERIMENTAL TECHNIQUES .......................... 661
9.1 HOPKINSON BAR, KOLSKY BAR, AND RECOVERY SYSTEMS .......... 663
9.1.1 Historical Origin of Hopkinson Techniques ......... 663
9.1.2 Classical Kolsky Method ........................... 664
9.1.3 Limitation of the Kolsky Method ................... 665
9.2 MOMENTUM TRAPPING FOR TENSION AND COMPRESSION HOPKINSON
BARS ..................................................... 666
9.2.1 Tension Hopkinson Bar with Momentum Traps ......... 666
9.2.2 Stress Reversal Technique ......................... 669
9.2.3 Compression Hopkinson Bar with Momentum Trap ...... 670
9.2.4 Strain-rate Jump .................................. 671
9.2.5 Strain-rate Jump in Tension ....................... 674
9.2.6 Recovery After Combined Compression-tension
Loading ........................................... 675
9.2.7 Recovery After Tension and Compression Loading .... 677
9.2.8 Stress-pulse Profile Control ...................... 678
9.3 HIGH-TEMPERATURE DYNAMIC RECOVERY EXPERIMENTS ............ 679
9.3.1 High-temperature, High Strain-rate Compression
Testing of Metals ................................. 679
9.3.2 Isothermal Flow Stress of Metals at High
Temperatures and High Strain Rates ................ 681
9.3.3 Temperature Rise During High Strain-rate Plastic
Deformation ....................................... 683
9.3.4 Interrupted Hopkinson Experiments and Typical
Results ........................................... 684
Infrared Detector Calibration ............................ 686
Results and Discussion ................................... 687
Elastic Energy of Dislocations ........................... 689
9.4 HOPKINSON TECHNIQUES FOR DYNAMIC TRIAXIAL COMPRESSION
TESTS .................................................... 690
9.4.1 Operation of Triaxial Hopkinson System ............. 691
9.5 SPECIAL TRIAXIAL CELL TO TEST FRICTIONAL GRANULES ........ 692
9.5.1 Specimen Preparation and Installation ............. 693
9.5.2 Experimental Procedure ............................ 696
9.5.3 Experimental Control and Data Acquisition ......... 699
9.5.4 Typical Experimental Results ...................... 699
9.5.5 Energy Dissipation and Pore Water Pressure ........ 702
9.6 RADIOGRAPHIC OBSERVATION OF SHEARBANDS IN LIQUEFACTION ... 703
9.6.1 Experimental Setup ................................ 705
9.6.2 Experimental Observation .......................... 707
9.7 REFERENCES ............................................... 711
CITED AUTHORS ................................................. 715
SUBJECT INDEX ................................................. 723
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